Asymptotic behavior of the estimation error covariance of quaternion estimators

Avishy Carmi; Yaakov Oshman

doi:10.2514/1.35781

Asymptotic behavior of the estimation error covariance of quaternion estimators

Avishy Carmi, Yaakov Oshman

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.

Original language	English
Pages (from-to)	1665-1676
Number of pages	12
Journal	Journal of Guidance, Control, and Dynamics
Volume	31
Issue number	6
DOIs	https://doi.org/10.2514/1.35781
State	Published - 1 Jan 2008
Externally published	Yes

ASJC Scopus subject areas

Control and Systems Engineering
Aerospace Engineering
Space and Planetary Science
Electrical and Electronic Engineering
Applied Mathematics

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10.2514/1.35781

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title = "Asymptotic behavior of the estimation error covariance of quaternion estimators",

abstract = "The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.",

author = "Avishy Carmi and Yaakov Oshman",

note = "Funding Information: This research was supported by the Israel Science Foundation under grant no. 1032/04 and by the Technion–Israel Institute of Technology{\textquoteright}s Asher Space Research Fund. The authors thank John L. Junkins for his comment that motivated Sec. IV and F. Landis Markley for helpful discussions.",

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AU - Oshman, Yaakov

N1 - Funding Information: This research was supported by the Israel Science Foundation under grant no. 1032/04 and by the Technion–Israel Institute of Technology’s Asher Space Research Fund. The authors thank John L. Junkins for his comment that motivated Sec. IV and F. Landis Markley for helpful discussions.

PY - 2008/1/1

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N2 - The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.

AB - The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.

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Asymptotic behavior of the estimation error covariance of quaternion estimators

Abstract

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